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Properties

Label 2-735-105.32-c1-0-5
Degree $2$
Conductor $735$
Sign $-0.202 - 0.979i$
Analytic cond. $5.86900$
Root an. cond. $2.42260$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.298 − 0.0799i)2-s + (−1.15 + 1.29i)3-s + (−1.64 − 0.952i)4-s + (−1.56 − 1.59i)5-s + (0.447 − 0.292i)6-s + (0.852 + 0.852i)8-s + (−0.333 − 2.98i)9-s + (0.340 + 0.600i)10-s + (−0.660 − 0.381i)11-s + (3.13 − 1.02i)12-s + (2.27 − 2.27i)13-s + (3.86 − 0.184i)15-s + (1.71 + 2.97i)16-s + (−1.25 − 4.69i)17-s + (−0.138 + 0.916i)18-s + (−1.41 + 0.818i)19-s + ⋯
L(s)  = 1  + (−0.210 − 0.0565i)2-s + (−0.666 + 0.745i)3-s + (−0.824 − 0.476i)4-s + (−0.701 − 0.712i)5-s + (0.182 − 0.119i)6-s + (0.301 + 0.301i)8-s + (−0.111 − 0.993i)9-s + (0.107 + 0.189i)10-s + (−0.199 − 0.114i)11-s + (0.904 − 0.297i)12-s + (0.629 − 0.629i)13-s + (0.998 − 0.0476i)15-s + (0.429 + 0.744i)16-s + (−0.305 − 1.13i)17-s + (−0.0327 + 0.215i)18-s + (−0.325 + 0.187i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.202 - 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.202 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.202 - 0.979i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ -0.202 - 0.979i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.198901 + 0.244350i\)
\(L(\frac12)\) \(\approx\) \(0.198901 + 0.244350i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.15 - 1.29i)T \)
5 \( 1 + (1.56 + 1.59i)T \)
7 \( 1 \)
good2 \( 1 + (0.298 + 0.0799i)T + (1.73 + i)T^{2} \)
11 \( 1 + (0.660 + 0.381i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.27 + 2.27i)T - 13iT^{2} \)
17 \( 1 + (1.25 + 4.69i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (1.41 - 0.818i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.98 - 7.39i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + 4.94T + 29T^{2} \)
31 \( 1 + (2.96 - 5.13i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.915 - 3.41i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 - 4.35iT - 41T^{2} \)
43 \( 1 + (-2.69 + 2.69i)T - 43iT^{2} \)
47 \( 1 + (-4.14 - 1.10i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-6.71 + 1.79i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (3.84 - 6.65i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.19 - 3.80i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.0471 - 0.0126i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 12.4iT - 71T^{2} \)
73 \( 1 + (0.359 + 1.34i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (3.66 - 2.11i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.05 - 5.05i)T + 83iT^{2} \)
89 \( 1 + (-0.453 - 0.785i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.73 + 3.73i)T + 97iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.61352502132510994089677511409, −9.690119881536472834717114567137, −9.063723725466167523674154307831, −8.323180148042619947956391485403, −7.24274450375559980702335259415, −5.72452781138042768138765925156, −5.27380697675092261032980750095, −4.30787248486233450796615286866, −3.49105624118477827240729120904, −1.08135550000188241768396255583, 0.23940109493312316687464667811, 2.17918296001879295831492096878, 3.77330497965977160688006167400, 4.49933386329592019428261084026, 5.88509698407966088897962758723, 6.72040888102424598890730143420, 7.59395550276703606293941619891, 8.254461897381921006588642754224, 9.072302267306423363136586096677, 10.44021781446724808517928541989

Graph of the $Z$-function along the critical line